Technology has recently been developed which offers an excellent opportunity to design systems with the ability to help people in their own houses. In particular, assisting elderly people in their environments is something that can significantly improve their quality of life. However, helping elderly people outside their usual environment is also necessary, to help them to carry out daily tasks like shopping. In this paper we present a person-following shopping cart assistance robot, capable of helping elderly people to carry products in a supermarket. First of all, the paper presents a survey of related systems that perform this task, using different approaches, such as attachable modules and computer vision. After that, the paper describes in detail the proposed system and its main features. The cart uses ultrasonic sensors and radio signals to provide a simple and effective person localization and following method. Moreover, the cart can be connected to a portable device like a smartphone or tablet, thus providing ease of use to the end user. The prototype has been tested in a grocery store, while simulations have been done to analyse its scalability in larger spaces where multiple robots could coexist.